Plane wave expansion method (PWE) refers to a computational technique in electromagnetics to solve the Maxwell's equations by formulating an eigenvalue problem out of the equation. This method is popular among the photonic crystal community as a method of solving for the band structure (dispersion relation) of specific photonic crystal geometries. PWE is traceable to the analytical formulations, and is useful in calculating modal solutions of Maxwell's equations over an inhomogeneous or periodic geometry. It is specifically tuned to solve problems in a time-harmonic forms, with non-dispersive media (a reformulation of the method named Inverse dispersion allows frequency-dependent refractive indices). Various open-source implementations of plane wave expansion method for calculating band diagrams were published in the photonics literature, including MPB.
Principles Plane waves are solutions to the homogeneous Helmholtz equation, and form a basis to represent fields in the periodic media. PWE as applied to photonic crystals as described is primarily sourced from Dr. Danner's tutorial. The electric or magnetic fields are expanded for each field component in terms of the Fourier series components along the reciprocal lattice vector. Similarly, the dielectric permittivity (which is periodic along reciprocal lattice vector for photonic crystals) is also expanded through Fourier series components.
1 ϵ r = ∑ m = − ∞ + ∞ K m ϵ r e − i G ⋅ r {\displaystyle {\frac {1}{\epsilon _{r}}}=\sum _{m=-\infty }^{+\infty }K_{m}^{\epsilon _{r}}e^{-i\mathbf {G} \cdot \mathbf {r} }}
E ( ω , r ) = ∑ n = − ∞ + ∞ K n E y e − i G ⋅ r e − i k ⋅ r {\displaystyle E(\omega ,\mathbf {r} )=\sum _{n=-\infty }^{+\infty }K_{n}^{E_{y}}e^{-i\mathbf {G} \cdot \mathbf {r} }e^{-i\mathbf {k} \cdot \mathbf {r} }}
with the Fourier series coefficients being the K numbers subscripted by m, n respectively, and the reciprocal lattice vector given by G {\displaystyle \mathbf {G} } . In real modeling, the range of components considered will be reduced to just ± N max {\displaystyle \pm N_{\max }} instead of the ideal, infinite wave. Using these expansions in any of the curl-curl relations like,
1 ϵ ( r ) ∇ × ∇ × E ( r , ω ) = ( ω c ) 2 E ( r , ω ) {\displaystyle {\frac {1}{\epsilon (\mathbf {r} )}}\nabla \times \nabla \times E(\mathbf {r} ,\omega )=\left({\frac {\omega }{c}}\right)^{2}E(\mathbf {r} ,\omega )}
and simplifying under assumptions of a source free, linear, and non-dispersive region we obtain the eigenvalue relations which can be solved.
Example for 1D case
For a y-polarized z-propagating electric wave, incident on a 1D-DBR periodic in only z-direction and homogeneous along x,y, with a lattice period of a. We then have the following simplified relations:
1 ϵ r = ∑ m = − ∞ + ∞ K m ϵ r e − i 2 π m a z {\displaystyle {\frac {1}{\epsilon _{r}}}=\sum _{m=-\infty }^{+\infty }K_{m}^{\epsilon _{r}}e^{-i{\frac {2\pi m}{a}}z}}
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